How derivatives affect the shape of a graph

Exponential Growth

If a population has a constant birth rate through time and is never limited by food or disease, it has what is known as exponential growth. With exponential growth the birth rate alone controls how fast (or slow) the population grows.

Click the following button to run an applet you can use to experiment with exponential growth. If you are accessing this lesson over a slower network connection it may take several seconds for the applet to appear.

You need Java installed and enabled to run this

The applet initially shows a habitat containing of two fish. Near the top of the window on the left it shows what generation we are in and on the right the population size is shown. Underneath the habitat view is an area where you can enter the average population birth rate. This rate is initialized to 1.5.

Near the bottom of the window are four buttons used to control the simulation. By clicking the Step button you can have the population "step" through one generation of time and see how many individuals are in the population the next generation. Clicking the Run button will automatically step a generation every second and the button will change to say Stop. Using either button try stepping through 20 generations and observe the results.

As you can see, the population rapidly gets crowded. You can see a graph of the population through time by clicking on the View Graph tab at the top of the applet. By enlarging the window you may be able to see more of the graph. At any time in this lesson you can switch between the Habitat view and the Graph view by clicking the appropriate tabs.

The two other buttons at the bottom are Reset and Reset All. Clicking either one will reset the population size to the initial value of 2 and set the generation to 0. The difference between the two buttons is apparent when viewing the graph. Reset will leave the previous results displayed and simply overlay them with new results. Reset All will clear the current and all previous graphs.

Take a few minutes now to become familiar with the applet. Using either

the Habitat or the Graph view try entering different birth rates and watch the population grow as you step through the generations (be sure to hit Reset or Reset All after changing the birth rate).

Experiment 1 :

In this experiment and those following you should leave the applet in the Graph view to help interpret the results.

Click Reset All to clear any past data. Now do a series of simulations using birth rates of 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0. Step or Run the population for 30 to 40 generations, or until the graph goes off the top of the window. Be sure to click the Reset button between each simulation in order to start over but leave the past results in place. Do this now.

As you can see, with an average birth rate of 1.0, the population didn't grow at all. It remained 2 the whole time. This is because each individual is just replacing itself the next generation.

With a birth rate of 1.2 the population grows very slowly at first and then starts to increase more quickly as it gets larger. This is a fundamental feature of exponential growth. The larger the population the more it appears to grow each generation.

The higher birth rates show the population increasing in size even faster. It is surprising how much difference a small change in the birth rate can make.

Experiment 2 :

Click Reset All to clear any past data. Set the average birth rate to be 1.5 and Step the population through 15 generations. Now without clicking either reset button change the birth rate to 0.8. This change means that each individual is now only producing (on average) 0.8 individuals in the next generation. You would expect the population size to decline. Step through 20 or 30 more generations and observe the results.

Initially the population declines quickly and then the curve flattens out as the population becomes small. This is similar to when the population was growing -- the most dramatic change occurs when the population is large. If you step the population out far enough the population number will go to zero and the population will become extinct.